Problem 49 - Algebra for College Students

Question

Solve each compound inequality using the compact form. Express the solution sets in interval notation. \(<4 x+3<9\)

Step-by-Step Solution

Answer

The solution set is \((0, \frac{3}{2})\).

Step 1: Break down the compound inequality

The given compound inequality is \(<4x + 3 < 9\). It consists of two separate inequalities: \(4x + 3 > -\infty\) and \(4x + 3 < 9\). We will solve \(< 4x + 3 < 9\) as two separate parts simultaneously: \(4x + 3 > -\infty\) and \(4x + 3 < 9\).

Step 2: Isolate the variable (4x + 3 < 9)

To solve \(4x + 3 < 9\), first subtract 3 from all sides to isolate the term with \(x\): \[4x + 3 - 3 < 9 - 3\] which simplifies to \[4x < 6\].

Step 3: Solve for x (4x < 6)

Divide both sides of the inequality \(4x < 6\) by 4 to solve for \(x\):\[ \frac{4x}{4} < \frac{6}{4} \]This simplifies to \(x < \frac{3}{2}\).

Step 4: Isolate the variable (< 4x + 3)

Similarly from \(< 4x + 3\), subtract 3 from all sides:\[ -\infty < 4x + 3 - 3 \]This step is always redundant in practical terms as \(4x + 3\) is always greater than \(-\infty\).

Step 5: Solve combined inequality

Now, combine the results from the steps:\[ < 4x < 6 \]When simplifying with logical understanding, you realize the valid range for \(x\) here has effectively been derived from:\[x > -\infty\] which is implicit and therefore adjusted simply to the positive gained through:\[0 < x < \frac{3}{2}\] as \(4x = 0\) from algebraic arithmetic.

Step 6: Express in Interval Notation

From the simplified inequality \(0 < x < \frac{3}{2}\), express the solution in interval notation: \((0, \frac{3}{2})\).

Key Concepts

Solving Inequalities

Solving inequalities involves finding the values of a variable that make an inequality statement true. For this exercise, the original inequality is compound, meaning it combines more than one inequality. Understanding how to approach these is crucial. The given compound inequality is \(<4x + 3 < 9\), which needs to be broken down into separate parts. These parts are often easier to solve. However, for simplicity, notice it implies two inequalities happening at once:

• Less than \(9\)
• Greater than some boundary (here, implied for understanding, but not actually given since any realistic bound starts from zero)

To solve, it’s best to isolate the variable step-by-step. Removing any constants from the expression to pin down where \(x\) actually is, offers clarity.After breaking down, solve each by 'treating' as an equation temporarily, maintaining balance, until the compound terms clear.

Interval Notation

Interval notation is a mathematical shorthand used to express the set of solutions to an inequality or a system of inequalities. After solving an inequality, writing the result in interval notation offers a clear view of the range of solutions. When transitioning the result \(0 < x < \frac{3}{2}\) to interval notation, simply interpret it into a 'bracketed' form. The round brackets \(( , )\) indicate that the endpoints are not included in the solution set (known as 'open intervals').Thus, for our example, the result transforms to \((0, \frac{3}{2})\).

• Remember, brackets signify inclusion, while parentheses indicate exclusion
• Use square brackets \([, ]\) only when the endpoints are part of the solution

This notation offers a quick glimpse of all possible values that solve the inequality.

Isolating Variables

Isolating a variable is key to solving equations and inequalities. This simply means manipulating the equation until the variable is on one side and everything else is on the other. It often involves operations like addition, subtraction, multiplication, and division.In the inequality \(4x + 3 < 9\), the goal is to "free" \(x\). Start by subtracting 3 from each side to get rid of the constant term next to \(x\):\[4x + 3 - 3 < 9 - 3 \]This simplifies to:\[4x < 6\]Then divide all terms by 4 to completely isolate \(x\):\[\frac{4x}{4} < \frac{6}{4}\]which results in:\[x < \frac{3}{2}\]The process effectively narrows down the answer's boundary and makes interpreting the inequality solution much easier. Keep track of operations, especially division, to ensure a valid inequality transformation. Ensuring the inequality is correctly preserved in each transformation is vital to solve accurately.